The present invention relates to digital filters, and in particular to an Infinite Impulse Response (IIR) digital filter for filtering a sequence of discrete samples of a digital signal with low coefficient sensitivity.
Digital recording and playback systems for processing a digital signal of equally spaced pulse samples such as pulse-code modulated (PCM) signals usually make use of attenuators and equalizers comprising digital filters for purposes of modifying the signal level and frequency response characteristic of the system. Digital filters are broadly divided into Infinite Impulse Response digital filters and Finite Impulse Response (FIR) digital filters. While the FIR digital filter has no stability problems due to the absence of a feedback loop and provides a linear phase characteristic, complex circuits result due to the large value of filter order. On the other hand, the IIR digital filter can be constructed with a filter order ten times less than the FIR filter although the phase linearity is not guaranteed.
IIR digital filters are classified into direct form and canonical form according to their internal connections and structures which are respectively shown in FIGS. 1a and 1b in which a sequence of input discrete samples x.sub.n at time nT is translated into a sequence of output discrete samples y.sub.n at time nT (where T is the sampling interval). The direct form IIR digital filter of FIG. 1a comprises a feedforward circuit including cascaded delay circuits each providing a unit delay time represented by Z.sup.-1 and an multiplier having a coefficient a.sub.2 for multiplying the delayed input samples x.sub.n by a coefficient a.sub.2. Multipliers having coefficients a.sub.0 and a.sub.1 are respectively coupled from the inputs of the feedforward delay circuits to an adder. A feedback circuit is provided comprising cascaded feedback delay circuits each providing a unit delay time to the output samples from the adder and multipliers having coefficients -b.sub.1 and -b.sub.2 respectively coupled from the outputs of the feedback delay circuits to the adder. The canonical IIR digital filter, FIG. 1b, comprises a pair of first and second adders and a pair of multipliers coupled in series between input and output terminals. A pair of feedback circuits including a pair of unit delay circuits in a common path connected from the output of the first adder and multipliers is provided to apply feedback signals to the inputs of the first adder. A pair of feedforward circuits including the common delay circuits and a pair of multipliers is coupled from the output of the first adder to the inputs of the second adder.
These types of digital filter have an inherent limitation in coefficient wordlength and require that the poles and zeros of the transfer function must be shifted for finite length quantization as in the case of FIR digital filters. For this reason, the coefficient quantization results in a deviation of the actual transfer function from desired values, thus resulting in the deviation of gain and frequency response characteristic from those desired. According to the description of "Digital Processing of Signals", McGraw-Hill, 1969 by B. Gold and C. M. Rader, the characteristics of a digital equalizer is highly sensitive when the allowable root locations have low density distribution in the z-plane as indicated by a hatched area I in FIG. 3, if the center frequency of the filter is low and the Q value is high.
If it is assumed that a digital equalizer is constructed using the IIR digital filter of FIG. 1a in a manner as shown and described in Japanese Patent Publication (Tokkaisho) 56-120211 filed by the same applicant as the present invention (with the center frequency fo=20 Hz, Q=3, signal level L=12 dB at fo with a sampling frequency 44.0569 kHz), frequency response characteristics as shown in FIG. 2 will be obtained for different wordlengths. As is evident from FIG. 2, the curves obtained for wordlengths larger than 25 correspond to design values, while other curves for smaller coefficient wordlengths increasingly deviate from the ideal with decrease in wordlength.
Although small coefficient wordlength is favored for economy purposes, this results in the undesirable circumstances just described. On the other hand, a longer wordlength would require many multipliers. Various attempts have thus far been made to reduce the coefficient sensitivity of the digital filter.
One typical prior art approach is described in "A Proposal to Find Suitable Canonical Structures for the Implementation of Digital Filters with Small Coefficients Wordlength", NTZ, 25, 8, pages 377-382, 1972, by E. Avenhouse. This method is based on the fact that the root positions have a uniform density distribution entirely in the z-plane. However, the disadvantages are a high degree of freedom of implementation and a greater number of multiplication operations than direct form digital filters.
A second approach to small coefficient sensitivity is proposed by R. G. Agarwal and C. S. Burrus in "New Recursive Digital Filter Structures Having Very Low Density Sensitivity and Roundoff Noise", IEEE Transactions, CAS-22 (December 1975). This method is concerned with digital filters of second-order low coefficient sensitivity and proposes two alternatives for narrow bandwidth filters in which the sensitivity problem is particularly severe. According to one such alternative the poles of the transfer function H(z.sup.-1) given by Equation 1 are represented by coefficients given in Equation 2 as follows: ##EQU1## where, r=radial distance of z-plane poles from the point of origin and 0=angle of the poles to the real axis.
Being an approximation, this method lacks accuracy and has the disadvantage of high degree of implementation freedom. The second alternative involves substituting 1--z.sup.-1 for z.sup.-1 in Equation 1 in order to provide small coefficient sensitivity in the low frequency range of the spectrum while having high coefficient sensitivity in the high frequency range. Since small coefficient sensitivity could not be obtained, this alternative method is also unsatisfactory for requirements that small coefficient sensitivity be effective over the full frequency range of the spectrum.
A third approach to small coefficient sensitivity is proposed by A. Nishihara in "Low-Sensivity Digital Filters with a Minimal Number of Multipliers", described in a technical journal published by the Institute of Electronics and Communication Engineers of Japan, September 1978 and A. Nishihara and Y. Moriyama in "Minimization of Sensitivities in Digital Filters by Coefficient Conversion", described in an August 1968 journal of the IECE. Similar to the Avenhouse method, this approach is concerned with narrow bandwidth filter structures where the density of allowable root locations of the bandwidth does not decrease in areas close to z=1 to a level lower than the density for other frequencies. However, the problem is a high degree of implementation freedom, which is disadvantageous for universal applications. Furthermore, the coefficient sensivity is evaluated in terms of amplitude sensitivity. Since the latter is variable as a function of frequency it is difficult to determine at which frequency the sensitivity evaluation is to be made.